Heights and a circle

[ins-]

In the acute-angled triangle $ABC$ $AA'$, $BB'$, $CC'$ are the heights through the vertices $A$, $B$, $C$. $k$ is a circle through $A$ and $B$ intersecting $BC$ and $AC$ at the points $A_1$ and $B_1$ respectively and $AA'$ and $BB'$ at the points $A_2$ and $B_2$. Prove that $A_1A_2=B_1B_2$ and $A_1A_2$, $B_1B_2$, $CC'$ intersects at a common point.

(Оффтоп)

It is very easy problem. Even I managed to solve it